Helical surfaces with a constant ratio of principal curvatures

Yang Liu, Olimjoni Pirahmad, Hui Wang, Dominik L. Michels, Helmut Pottmann*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We determine all helical surfaces in three-dimensional Euclidean space which possess a constant ratio a: = κ1/ κ2 of principal curvatures (CRPC surfaces), thus providing the first explicit CRPC surfaces beyond the known rotational ones. Our approach is based on the involution of conjugate surface tangents and on well chosen generating profiles such that the characterizing differential equation is sufficiently simple to be solved explicitly. We analyze the resulting surfaces, their behavior at singularities that occur for a> 0 , and provide an overview of the possible shapes.

Original languageEnglish (US)
Pages (from-to)1087-1105
Number of pages19
JournalBeitrage zur Algebra und Geometrie
Issue number4
StatePublished - Dec 2023

Bibliographical note

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  • Helical surface
  • Surface with a constant ratio of principal curvatures
  • Weingarten surface

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology


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